Subgroups — #28383685

Order 14272.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813160	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₄₄₆	C₂	1	1	yes	maximal, characteristic
3	#20297235	C₂	1	1	yes	maximal, characteristic
4	#20297228	C₂	1	1	yes	maximal, characteristic
5	#20297227	C₂	1	1	yes	maximal, characteristic
6	#20297215	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
9	#16813160	C₄	1	4	yes	complemented
10	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
13	Dic₂₂₃⋊C₄	—	2	1	no	—
14	C₄×Dic₂₂₃	—	2	1	no	—
15	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₂₂₃	—	2	1	no	—
17	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₄²×C₂₂₃	—	2	1	no	—
19	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
20	Dic₄₄₆	—	2	1	no	—
21	Dic₄₄₆	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₂₃	—	2	1	no	—
23	C₂×Dic₂₂₃	—	2	1	no	—
24	C₂₂₃⋊C₈	—	2	1	no	—
25	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
26	Q₈×C₂₂₃	—	2	1	no	—
27	Q₈×C₂₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₂₂₃	—	2	1	no	—
29	C₂×C₄×C₂₂₃	—	2	1	no	—
30	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₂₃	—	2	2	no	—
35	C₄×C₂₂₃	—	2	1	no	—
36	C₄×C₂₂₃	—	2	1	no	—
37	C₄×C₂₂₃	—	4	1	no	—
38	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₂₂₃	#41	1	1	yes	characteristic
40	Dic₂₂₃	—	2	2	no	—
41	Dic₂₂₃	—	4	1	no	—
42	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
43	C₂×C₂₂₃	#137	1	1	yes	characteristic
44	C₂×C₂₂₃	#137	1	1	yes	characteristic
45	C₂₂₃	#441	1	1	yes	characteristic, complemented, Sylow
46	#441	—	223	1	no	maximal, Sylow
47	C₂×Q₁₆	—	223	1	no	—
48	C₄×Q₈	—	223	1	no	—
49	C₄×Q₈	D₂₂₃	1	1	yes	characteristic
50	#106	—	223	1	no	—
51	#103	—	223	1	no	—
52	#103	—	223	1	no	—
53	#97	—	223	1	no	—
54	C₂×Q₈	—	223	1	no	—
55	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
56	Q₁₆	—	223	4	no	—
57	C₂×C₈	—	223	1	no	—
58	C₂×C₈	—	223	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	446	1	no	—
61	C₄⋊C₄	—	223	1	no	—
62	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
63	C₄²	—	2	1	no	—
64	C₄²	—	446	1	no	—
65	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
66	Q₈	—	2	1	no	—
67	Q₈	—	446	1	no	—
68	Q₈	—	223	2	no	—
69	Q₈	C₄×D₂₂₃	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	446	1	no	—
73	C₂×C₄	—	446	1	no	—
74	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
77	C₈	—	446	1	no	—
78	C₈	—	223	2	no	—
79	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	892	1	no	—
85	C₄	—	446	2	no	—
86	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
87	C₄	#16813170	1	1	yes	characteristic
88	C₂	C₄×C₂²⋊D₂₂₃	1	1	yes	characteristic
89	C₂	#20297334	1	1	yes	characteristic
90	C₂	#20297335	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄×Q₈×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₄×Q₈	D₂₂₃
hypocenter	C₂₂₃	#441
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. #16813160	C₄	semidirect
45. C₂₂₃	#441	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16813160	2
C₄×Dic₄₄₆	2
#20297235	2
#20297228	2
#20297227	2
#20297215	2
C₄×Q₈×C₂₂₃	2
#441	223